Quantitative analysis of signal related measurements for trending and pattern recognition

ABSTRACT

A computerized method for quantitative analysis of signal related measurements, performed with one or more processors, is disclosed. An estimated signature typifying a characteristic feature of the signal related measurements is produced. Multidimensional statistics on the signal related measurements are computed in a multidimensional space with respect to the estimated signature. Matching likelihoods of the signal related measurements are quantified based on distances of the signal related measurements with respect to a shell manifold derived from the multidimensional statistics and enveloping a signature manifold in the multidimensional space. Multidimensional statistics on the estimated signature and trending and pattern recognition are also possible from the signal multidimensional projection.

FIELD OF THE INVENTION

The invention relates to a computerized method for quantitative analysisof signal related measurements, for example for trending and patternrecognition to monitor an operating state of an apparatus or equipment.

BACKGROUND

Most of trending and matching algorithms consider additive noise at thefirst level: the measurement is processed one-dimensionally where thenoise is a random amount added for each signal sample, instead ofprocessing measurements as a sum of signal vector and noise vector.Better results are obtained when considering a statistical modelling ofsignal and noise. A deeper step is a process modeling which allows amonitoring of internal process parameters.

WO 2012/162825 (Léonard) proposes a dynamic time clustering involvingstatic signature discrimination using a projection of the measurement asthe sum of signal and noise vectors in a multidimensional space

^(N). However, quantitative analysis of the signals is limited andpattern recognition of a signature and moving signature conditions arenot addressed.

In the field of process trending, many monitoring algorithms useenvelopes (continuous) or bands (discrete) alarms that generate an alarmwhen exceeded. Manual adjustment appears difficult and inaccurate.Self-adjustment of alarm levels is greatly desired to increaserobustness and sensitivity and to facilitate implementation. Mostly, themethods only consider the greatest excess to raise an alarm. The alarmlevel is typically set high in order to avoid “false positive”indications generated by random noise. This setting reduces thesensitivity and increases the probability of “false negative”indications (defect missed). Moreover, several small differences belowthe alarm level can be as problematic as one large excursion on a smallportion outside the alarm limit “False positive” indications in trendingand pattern recognition appear to be a great concern. The followingpatents only partially overcome these problems. U.S. Pat. No. 8,239,170(Wegerich) proposes a method for detection of state changes, or forsignature recognition and classification. Wavelet analysis, frequencyband filtering or other methods may be used to decompose the signal intocomponents. The decomposed signal is compared with a library signature.The recognized signature indicates data being carried in the signal.U.S. Pat. No. 6,522,978 (Chen et al.) proposes a method for predictingweb breaks in a paper machine Principal components analysis andclassification and regression tree modeling are used to predict webbreak sensitivity from measurements. U.S. Pat. No. 6,278,962(Klimasauskas et al.) proposes a hybrid analyzer using a linear modelwith a trained neural network model to predict the process variables.U.S. Pat. No. 4,937,763 (Mott) proposes the monitoring of amulti-variable process by comparing observations acquired when thesystem is running in an acceptable state with current observationsacquired at periodic intervals thereafter to determine if the process iscurrently running in an acceptable state. U.S. Pat. No. 8,250,006 (Smitset al.) proposes a predictive algorithm using a genetic programmingtechnique that evolves a population of candidate algorithms throughmultiple generations. The predictive algorithm may be implemented in aninferential sensor that is used to monitor a physical, chemical, orbiological process, such as an industrial process in an industrialplant.

In the field of image processing, U.S. Pat. No. 5,253,070 (Hong)proposes a hardware circuit for automatically detecting a variation ofvideo information where the currently inputted video data is comparewith the stored video data to detect a portion different from each otheras the variation of video information. U.S. Pat. No. 4,081,830 (Mick etal.) proposes a motion and intrusion detection system which storesinformation concerning fixed scanned points. During subsequent scans,information concerning the respective scanned points is compared withprevious scans and threshold conditions are set up in order to detect analarm condition. US 2002/0057840 (Belmares, Robert J.) proposes a methodfor monitoring a field of view for visible changes using image digitalprocessing. U.S. Pat. No. 8,224,029 (Saptharishi et al.) proposes acamera system comprising an image capturing device, an object detectionmodule, an object tracking module and a match classifier. The matchclassifier determines whether the selected object image signaturematches the first object image signature. A training processautomatically configures the match classifier.

In the field of identification of molecular structures, U.S. Pat. No.7,606,403 (Haussecker et al.) proposes a capture of a plurality ofimages of one or more subjects using different imaging techniques,followed by parameters estimation from the plurality of images, usingone or more models of known molecular structures to provide amodel-based analysis. U.S. Pat. No. 8,200,440 (Hubbell et al.) proposesa method of analyzing data from processed images of biological probearrays where a cluster corresponds to different genotypes using aGaussian cluster model. U.S. Pat. No. 7,356,415 (Pitman et al.) proposesa method in a data processing system for generating and storing in adatabase descriptor vectors and reference frames for at least one regionof a molecule. For each particular subset of component vectors, themethod calculates a probability value for the F-distributed statisticassociated with the particular subset, identifies the subset ofcomponent vectors associated with the selected probability value andgenerates a mapping to a space corresponding to the subset. In U.S. Pat.No. 6,671,625 (Gulati), a dot spectrogram is analyzed using clusteringsoftware to generate a gene array amplitude pattern representative ofmutations of interest.

In the field of medical condition evaluation, U.S. Pat. No. 8,065,092(Khan et al.) proposes a method based on high experimental dimensionaldata using training and supervised pattern recognition to determine ifan unknown set of experimental data indicates a disease condition, apredilection for a disease condition, or a prognosis about a diseasecondition.

In the field of radar, U.S. Pat. No. 7,034,738 (Wang et al.) proposes amethod for classifying radar emitters by sorting multi-dimensionalsamples into a plurality of data clusters based on their respectiveproximity to the data clusters, each data cluster representing aclassification of a radar emitter.

In the field of financial predictions, US 2013/0031019 (Herzog; JamesPaul) proposes a monitoring system for determining the future behaviorof a financial system. An empirical model module is configured toreceive reference data that indicate the normal behavior of the system,and processes pattern arrays in order to generate estimate values basedon a calculation that uses an input pattern array and the reference datato determine a similarity measure.

SUMMARY

An object of the invention is to provide a computerized method forquantitative analysis of signal related measurements which addresses theabove shortcomings of the prior art.

Another object of the invention is to provide such a method which may beused for trending and pattern recognition and may achieve patternrecognition of a signature and handle moving signature conditions inorder to track a signature evolving for example in response to a changein the operating state of an apparatus or relating to a way in which thesignal measurements are taken.

Another object of the invention is to provide such a method which maycontinuously adjust alarm bounds and sensitivity and may considersimultaneously in a single criterion all deviations for all signalrelated measurements.

According to an aspect of the invention, there is provided acomputerized method for quantitative analysis of signal relatedmeasurements, the method comprising the steps of, performed with one ormore processors:

-   -   producing an estimated signature typifying a characteristic        feature of the signal related measurements;    -   computing multidimensional statistics on the signal related        measurements in a multidimensional space with respect to the        estimated signature; and    -   quantifying matching likelihoods of the signal related        measurements based on distances of the signal related        measurements with respect to a shell manifold derived from the        multidimensional statistics and enveloping a signature manifold        in the multidimensional space.

The method of the invention may be performed on a computer or amicrocontroller having one or more processors and other devices,peripherals and accessories such as one or more memories, an I/O card, adisplay, etc.

According to another aspect of the invention, there is provided a systemfor monitoring an operating condition of an apparatus, comprising:

-   -   a measuring arrangement connectable to the apparatus, the        measuring arrangement being configured to measure one or more        predetermined operating parameters of the apparatus and produce        signal related measurements thereof;    -   a memory having a statistics database;    -   a processor connected to the measuring arrangement and the        memory, the processor being configured to process the signal        related measurements, produce multidimensional statistics on the        signal related measurements and an estimated signature typifying        a characteristic feature of the signal related measurements,        updating the statistics database with the signal related        measurements and the multidimensional statistics, and produce        diagnosis data indicative of the operating condition of the        apparatus as function of matching likelihoods of the signal        related measurements quantified based on distances of the signal        related measurements with respect to a shell manifold derived        from the multidimensional statistics and enveloping a signature        manifold in the multidimensional space; and    -   an output unit connected to the processor for externally        reporting the diagnosis data.

BRIEF DESCRIPTION OF THE DRAWINGS

A detailed description of preferred embodiments will be given hereinbelow with reference to the following drawings:

FIG. 1 is a schematic diagram illustrating components and functions ofthe method according to an embodiment of the invention.

FIG. 2 is a schematic diagram illustrating a system monitoring operatingconditions of an apparatus according to an embodiment of the invention.

FIG. 3 is a schematic diagram providing a N-dimensional illustration ofa measurement cluster and a corresponding signature.

FIG. 4 is a schematic diagram illustrating a measurement X_(m), anestimated signature S′_(i) and their respective distance D_(m,i) withmeasurement dispersion and hypersphere radius deviation (HRD) amplitude.

FIG. 5 is a schematic diagram providing a N-dimensional illustration ofa trend of a signature at four different stages.

FIG. 6 is a schematic diagram providing a N-dimensional illustration ofa trending of a signature at two different stages where a measurementdispersion appears larger for a last signature S_(i+1).

FIG. 7 is a schematic diagram illustrating hyperspheres of twosignatures related to a same denoised signature and a hyperspherecorresponding to signature subtraction.

FIG. 8 is a schematic diagram illustrating details for

^(N) measurement and signature statistics according to the invention.

FIG. 9 is a schematic diagram illustrating an estimated signature S′_(i)to estimated signature S′_(j) distance D_(i,j) with total signaturedispersion σ_(i,j) ^(S−S) and hypersphere to signature distance (HSD).

FIG. 10 is a schematic diagram providing a N-dimensional illustration ofa distribution of measurement membership likelihoods for a set ofsignatures.

FIG. 11 is a schematic diagram providing a N-dimensional illustration ofa measurement cluster and a corresponding estimated signature S′_(i,x)at different evolution steps.

FIG. 12 is a graph illustrating concatenated time series of histogramsgenerated using vibroacoustic measurements.

FIG. 13 is a graph illustrating a curve resulting from application ofthe method according to the invention on the vibroacoustic measurementsof FIG. 12 and alarm settings.

FIG. 14 is a graph illustrating an estimated covariance matrix generatedusing the vibroacoustic measurements of FIG. 12.

FIG. 15 is a schematic diagram illustrating an exemplary embodiment ofthe invention where a sensor is moved in the vicinity of a dielectricdefect in a high-voltage junction.

FIGS. 16A, 16B, 16C, 16D, 16E and 16F are graphs illustrating a spatialevolution of a partial discharge signature derived from a measuring andprocessing of measurement samples from the sensor of FIG. 14 accordingto the method of the invention.

FIG. 17 is a schematic diagram providing a N-dimensional illustration ofa signature domain for a signature sensitive to the temperatureoperating condition.

FIG. 18 is a schematic diagram providing a N-dimensional cut viewillustration of the measurements dispersion around the signature domainwhere the dispersion amplitude increases when temperature decreases.

FIG. 19 is a schematic diagram providing a N-dimensional cut viewillustration of the shell manifold corresponding to measurementsprobability function density located in the neighbourhood of thesignature domain manifold.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As used in connection with this disclosure, the term “signal relatedmeasurements” refers to measurements derived from signals or samples ofsignals measured in respect with one or more like apparatuses (e.g. withsensors) or other kinds of signal sources and exhibiting possible noiseand a characteristic sequence repeatable from one measurement to anotherand defining a signature. Most of the non-repeatable part of themeasurements comes from the measured phenomenon intrinsic randomcontribution, ambient noise, the measurement system noise and thetemporal evolution of the phenomenon between the measurements. Themethod according to the invention enables detecting and quantifyingvariations of the measurements with respect to a signature, variationsof a signature with respect to an initial signature, and a matchinglikelihood of a signature or a measurement with a specific signaturepattern.

In an embodiment of the method according to the invention, a signalrelated measurement is projected in a multidimensional space

^(N) where noise appears as a single vector with a length and adirection for a given set of signal measurements. The method considers atotal noise dispersion in

^(N) instead of a sample-to-sample local noise dispersion. Projection ofa N samples signal in a multidimensional space

^(N) for several realizations of measurement gives a distribution closeto a hypersphere of radius r where r corresponds to a statisticalaverage of measurement dispersion. At a center of the hypersphere islocated an average signature. A boundary of the hypersphere has athickness called hardness. The measurement population, the measurementdispersion and the number N of dimensions fix a standard deviation ofthe signature and the hypersphere thickness. The signature andhypersphere thickness values are estimated by the method according tothe invention and used to scale up a deviation amplitude between themeasurements. In the case of a large number of dimensions, a measurementdistance with respect to the hypersphere shell density probabilityfunction converges to a Laplace-Gauss modeling and yields an analyticalstatistical formulation for a measurement likelihood.

In the method according to the invention, evolution of a signature canbe idle to slow, progressive or sudden. In an embodiment in respect withhigh voltage accessories partial discharge (PD) location in undergroundvault, displacement of a PD sensor shows a PD signature evolution asfast as the displacement is made. In such a case, if the sensor is movedback, the initial signature is found again. Pattern matching recognitionis used to compare each evolution step of the PD signature with areference signature corresponding to a known defect. In another PDdomain embodiment in respect with PD monitoring of high voltageequipment in a substation, the method of the invention may be used totrack an evolution of PD related to an insulation defect. FIG. 15 showsan exemplary embodiment of the invention in such a context. A sensor 20is moved in the vicinity of a dielectric defect in a high-voltagejunction 22. The sensor 20 may advantageously be formed of anelectromagnetic “sniffer” as disclosed in U.S. Pat. No. 8,126,664(Fournier et al.). A measurement sample 24 derived from signal measuringand processing from the sensor 20 at position 4 is shown in the figure.Positions 1 to 6 may for example correspond to a 3 cm (approximately)shift of the sensor 20 along a cable 26 connected to the junction 22.FIGS. 16A, 16B, 16C, 16D, 16E and 16F illustrate a spatial evolution ofa partial discharge signature derived from a processing of measurementsamples taken with the sensor 20 at respective positions 1, 2, 3, 4, 5and 6 (shown in FIG. 15) according to the method of the invention. Eachone of FIGS. 16A, 16B, 16C, 16D, 16E and 16F represents an evolutionstep of the signature. In another embodiment in respect withvibro-acoustic monitoring of equipments, the method of the invention maycompare a new vibro-acoustic measurement with a moving average (MA)signature and compare the MA signature with an initial signature. In anembodiment in respect with a rotating machinery, a measurement processedwith the method of the invention may be an order power spectrum of asignal produced by a sensor located on a rotating equipment or aconcatenation of a plurality of power spectrum corresponding todifferent locations on the same rotating equipment. In processmonitoring, e.g. chemical or manufacturing, the method of the inventionmay be used to monitor the process and diagnose emerging fault. In imageprocessing, a new image may be compared with an image signature todetect motion. If there is a database of signatures related to aconfirmed diagnosis, the method of the invention may be used for patternrecognition, e.g. match a current signature with a previous signaturestored in the database in order to provide a diagnostic and acorresponding likelihood.

Referring to FIG. 1, a measuring arrangement (or system) 101 producesone or more signals. As used in connection with this disclosure, theterms “measurement” and “signal related measurement” 103 represent aresult of a signal processing 102 applied on a signal as produced by themeasuring arrangement 101. For example, the measurement 103 may be atime series, an envelope, a power spectrum, a scalogram, a spectrogram,a 2D image, etc. Blocks (or modules) 104 and 105 respectively depictcomputation of multidimensional statistics on the estimated signature(or signature related data) and on the signal related measurements in amultidimensional space

^(N). In the

^(N) signature statistics 104, many measurements may be averaged togenerate the “estimated signature” and to generate some statisticalindicators characterizing a

^(N) cluster measurement dispersion. Many measurements may be related tomore than one estimated signature. The case of dynamically clusteringmeasurements to many signatures is treated in WO 2012/162825 (Léonard).When considering one signature 100 (as shown in FIG. 11) and manycorresponding measurements, the

^(N) measurement statistics 105 may be calculated for an individualmeasurement, usually the last measurement. Block (or module) 106 depictscomputation of a measurement trending from the

^(N) measurement statistics 105 and the

^(N) signature statistics 104. Block (or module) 108 depicts computationof measurement trending or pattern recognition from the

^(N) measurement statistics 105 and signature statistics that may bestored in a database as depicted by block 107. For example, when themeasurement trending 106 exhibits a significant deviation with respectto a signature, some pattern recognitions 108 are performed for a set ofsignatures stored in a database 107 that may be characteristic ofdocumented equipment failures. Block (or module) 109 depicts computationof signature trending or pattern recognition from the

^(N) signature statistics 104 and the database stored signaturestatistics 107. In the signature trending 109, a signature comparison iscomputed between the updated signature statistics 104 and a formersignature statistic 107. When the signature trending 109 exhibits asignificant deviation, some pattern recognitions 109 are performed for aset e.g. of signatures stored in the database. The trending and/orpattern recognition outputs of blocks 106, 108 and 109 may be used toproduce a diagnosis 116 which may take the form of diagnosis dataindicative for example of a current alarm state or other operatingconditions of an apparatus 118 (as shown in FIG. 2), possibly displayedand/or transmitted to other systems. The

^(N) signature statistics 104 and the

^(N) measurement statistics 105 are collected by the statistic databaselocal management module 112 which updates the statistic information inthe local database 132 and possibly transfers the information to otherdatabases. The local statistic database 132 may contain informationabout local cumulated measurements and information coming from systemswhich monitor similar apparatuses. For example, in the case of identicalapparatuses located in different countries which are subjected tovarious climates, merging the apparatus response for different operationtemperatures may be used to get a picture of the typical signature overa larger temperature range. The statistic database local management 112may share the statistic information with other databases 114. Thesharing can be enabled at predetermined time intervals or whensignificant new information is available.

The merging process of the apparatus responses cumulated by differentsystems must take into account the similarities between the apparatusresponses. Moreover, the cumulated measurements on a defective apparatusshould not be merged with a database of a healthy apparatus. One task ofthe statistic database local management 112 may be to select theappropriate information to complete an unknown part of the apparatusresponse statistics 104, 105. There is a compromise between a hole inthe local database and inconsistencies resulting from the merging ofsome inappropriate data.

Referring to FIG. 2, there is shown an embodiment of a system formonitoring an operating condition of an apparatus 118. A measuringarrangement 101 connectable to the apparatus 118 measures one or morepredetermined operating parameters of the apparatus 118 and producessignal related measurements from them. The measuring arrangement 101 mayhave one or more transducers 122 connected to a sampling unit 126through a signal conditioning unit 124. The signal conditioning unit 124may have an input for receiving operating condition signals 120 inrespect with the apparatus 118. The signal conditioning unit 124 may besuch as to accept analog and digital input signals and may includeelectrical protection, analog filtering, amplification and envelopdemodulation. The sampling unit 126 is intended to convert any analogsignal into a digital signal and may add time stamps to the sampledsignals which are then transmitted to a processor 128 (or manyprocessors interconnected and operating together) connected to themeasuring arrangement 101 and a memory 130 having a statistics database132. The

_(N) signature statistics 104, the

^(N) measurement statistics 105, the measurement trending 106, themeasurement trending or pattern recognition 108, the signature trendingor pattern recognition 109 and the statistical database local management112 may be embodied by the processor 128 in programmed form orotherwise, i.e. in electronic form depending on the design of theprocessor 128. The processor 128 may store and read data in the memory130 that contains the local statistics database 132. The processor 128is thus configured to process the signal related measurements, producemultidimensional statistics on the signal related measurements and anestimated signature typifying a characteristic feature of the signalrelated measurements, update the statistics database 132 with the signalrelated measurements and the multidimensional statistics, and producediagnosis data indicative of the operating condition of the apparatus118 as function of matching likelihoods of the signal relatedmeasurements quantified based on distances of the signal relatedmeasurements with respect to a shell manifold derived from themultidimensional statistics and enveloping a signature manifold in themultidimensional space, as it will be described with further detailshereinafter. The processor 128 may transmit the diagnosis data 116 to anoutput unit 136 such as a display or printer connected to it forexternally reporting the diagnosis data and to other systems monitoringlike apparatuses through a communication unit 134 connected to theprocessor 128 and the memory 132 and connectable to a communication link138 with the other systems for exchanging data with them. A control unit140 connected to the processor 128 and the measuring arrangement 101 andconnectable to the apparatus 118 may produce control signals for theapparatus 118 as function of control data produced by the processor 128based on the diagnosis data. The processor 128 may then have a controlmodule (in programmed or electronic form) for example producing a newcontrol setting point in response to alarm state data in the diagnosisdata 116, the new control setting point being transmitted to theapparatus 118 through the control unit 140. The alarm state data may bederived from the measurement trending, the measurement trending orpattern recognition and the signature trending or pattern recognitionmodules 106, 108, 109 (as shown in FIG. 1) and be indicative of anabnormal operating state of the apparatus 118 as it will be furtherdescribed hereinafter.

Referring to FIG. 3, for a signature S_(i) 100 (as shown in FIG. 11),assuming a repetitive pattern signature S_(i,n) over different noiserealizations where “n” is a subscript corresponding typically to time,frequency, wavelet scale or component order, let's consider a N samplesmeasurement

X _(mn) =S _(i,n) +n _(mn) , n∈[1, N]  (1)

where n_(mn) is the additive noise and “m” a subscript of themeasurement realization. Assuming a noise having a density probabilityfunction centered and many measurements 103 of a same S_(i,n) pattern, a

^(N) projection of the measurements exhibits a cluster centered on apoint of

^(N) corresponding to the “i” signature estimate 201

S′_(i)={S′_(i,1), S′_(i,2), . . . , S′_(i,N)}  (2)

where the measurements 103

X_(m)={X_(m,1), X_(m,2), . . . , X_(m,N)   (3)

are distributed close to a hypersphere shell 203. Among differentaveraging options, a signature S_(i) may be estimated from a Mmeasurements set using a uniform average

$\begin{matrix}{{S_{i} \cong S_{i}^{\prime}} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}\; {X_{m}.}}}} & (4)\end{matrix}$

This signature estimation forms a first part of the “

^(N) signature statistics” 104 (as shown in FIG. 1). In some trendingapplications, an updated estimated signature 201 and other estimationsare generated by a moving average process. Uniform average is presentedhere to keep the mathematical expressions short.

Referring to FIG. 4, for example for an Euclidian metric with uniformaveraging, the measurement to estimated signature distance 205

D _(m,i) =∥X _(m) −S′ _(i)∥=√{square root over (Σ_(n=1) ^(N)(X _(m,n)−S′ _(i,n))²)}  (5)

has an expected mean for a M measurements set. This set exhibits acluster centered on the estimated signature 201, not on the realsignature. The estimated signature 201 appears in eq. 5 for two reasons.First, the real signature is unknown and, second, an internal error ofthe signature estimation must not appear at this step. Using eq. 5, anexpected mean distance

$\begin{matrix}{{r_{i} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}\; D_{m,i}}} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}\; {\sqrt{\sum\limits_{n = 1}^{N}\; \left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}{for}\mspace{14mu} X_{m}}}} \in {{{cluster}\mspace{14mu}}^{''}i^{''}}}}},} & \left( {6a} \right)\end{matrix}$

or (possibly less accurate)

$\begin{matrix}{{r_{i} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}\; D_{m,i}^{2}}} = \; {{\sqrt{\underset{n = 1}{\overset{N}{\frac{1}{M}{\sum\limits_{m = 1}^{M}\sum}}}\; \left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}{for}\mspace{14mu} X_{m}} \in {{{cluster}\mspace{14mu}}^{''}i^{''}}}}},} & \left( {6b} \right)\end{matrix}$

appears close to the noise vector length expectation √{square root over(NE(n_(mn) ²))} where E ( ) is the expectation function. S′_(i,n) is aclosest point of the estimated signature with respect to X_(m,n), Nrepresenting a number of dimensions related to a representation form ofthe signal related measurements, M representing a number of signalrelated measurements X_(m) 103, and S′_(i) 201 representing theestimated signature produced in relation with a cluster i to which thesignal related measurements belong. This estimation (r_(i)) will bereferred to as measurement hypersphere radius 202. The double of theexpected measurement standard deviation 206 of the measurementdistribution relative to hypersphere shell calculated with

$\begin{matrix}{\sigma_{i}^{r} = \sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}\; \left( {r_{i} - \sqrt{\sum\limits_{n = 1}^{N}\; \left( {X_{m,n} - S_{i,n}} \right)^{2}}} \right)^{2}}}} & \left( {7a} \right)\end{matrix}$

from measurements or estimated with

$\begin{matrix}{\sigma_{i}^{r} = {{r_{i} \cdot \sqrt{\frac{1}{2\; N}}}\mspace{14mu} \left( {{Gaussian}\mspace{14mu} {white}\mspace{14mu} {noise}\mspace{14mu} {assumption}}\; \right)}} & \left( {7b} \right)\end{matrix}$

from the estimated radius length will be referred to as thickness 204(as shown in FIG. 3). In the case of Gaussian white noise, the estimateddispersion given with eq. 7b appears with less dispersion than thatcalculated with eq. 7a. The accuracy of the measurement standarddeviation 206 increases with the number M of measurement realizations.For a small M number, the estimation error on the standard deviation,i.e. the dispersion of measurement-to-shell dispersion 207

$\begin{matrix}{\left( \sigma_{i}^{\prime \; r} \right)^{2} = {{\frac{\sigma_{i}^{r\; 2}}{M - 1}\mspace{14mu} {or}\mspace{14mu} \sigma_{i}^{\prime \; r}} = {{r_{i} \cdot \sqrt{\frac{1}{2{N \cdot \left( {M - 1} \right)}}}}\mspace{14mu} \left( {{Gaussian}\mspace{14mu} {white}\mspace{14mu} {noise}\mspace{14mu} {assumption}}\; \right)}}} & (8)\end{matrix}$

should be also taken into account.

Referring back to FIG. 3, the measurement hypersphere radius r_(i) 202and shell thickness 2·σ_(i) ^(r) 204 are illustrated in

^(N). The

^(N) measurement statistics 105 (as shown in FIG. 1) include thehypersphere radius r_(i) 202, the measurement standard deviation σ_(i)^(r) 206 and the dispersion of measurement dispersion σ′_(i) ^(r) 207(as shown in FIG. 4).

The measurement hypersphere shell thickness 204 is function of themetric (e.g. Euclidian), the measurement signal-to-noise ratio (SNR) andthe number of time samples N. The ratio of the shell thickness 204 overthe hypersphere radius 202 tends to 0 when N→∞. This phenomenon iscalled sphere hardening. Calculated using numerous noise samples, theexpected distance ∥X_(m)−S′_(i)∥ appears constant and corresponds to themeasurement hypersphere radius 202.

The estimated measurement hypersphere radius r_(i) 202 is computed froma finite set on M measurement realizations. The estimated signatureS′_(i) 201 corresponds to the

^(N) coordinates that minimize the radius length (the estimatedhypersphere radius is an underestimation of the real radius length). Afourth part of the “

^(N) measurement statistics” 105 is formed of an estimated radius biaserror

$\begin{matrix}{{ɛ_{i}^{s} = {r_{i} \cdot \sqrt{\frac{1}{M\left( {M - 1} \right)}}}},{M > 1},} & (9)\end{matrix}$

that corresponds to an additional radius length observed for ameasurement which is not included in the computed estimated signature:{X_(m):m ∉ [1, M]}. This radius length bias is attributed to the averagesignature displacement when a new measurement is added to itsestimation.

Referring to FIG. 4, in a preferred trending analysis, a lastmeasurement is compared with a signature moving average and the movingaverage is compared with a reference signature established at the startof the monitoring. For the first comparison, the measurement is near thehypersphere shell and the statistical deviation is governed by a ratioof a measurement-to-hypersphere shell distance over a measurementstandard deviation 206 as expressed in eq. 7. Moreover, a dispersion ofmeasurement dispersion 207 is taken into account. FIG. 4 illustrates thedifferent contributions involved in a deviation probability estimation.The distance difference D_(m,i)−r_(i) when {X_(m):m ∈ [1, M]} orD_(m,i)−(r_(i)+ε_(i) ^(shu) ) when {X_(m):m ∉ [1, M]} is related to themeasurement total dispersion 209

(σ_(i) ^(X−S))²=(σ_(i) ^(r))² +(σ′_(i) ^(r))²   (10)

in order to obtain the probability density

$\begin{matrix}{{\Omega \left( {X_{m},S_{i}} \right)} = \frac{e^{{{- {({D_{m,i} - r_{i} - ɛ_{i}^{s}})}^{2}}/2}{(\sigma_{i}^{X - S})}^{2}}}{\sigma_{i}^{X - S}\sqrt{2\; \pi}}} & (11)\end{matrix}$

for a large number N of dimensions since for numerous errorscontribution, the hypersphere shell density probability functionconverges to a Laplace-Gauss modeling (central limit theorem).

When the dispersion function is unknown or does not correspond to apossible analytical modeling, a histogram build-up may be carried outwith the collected measurements to estimate the density probabilityfunction corresponding to the measurement population. A histograminterpolation may replace the modeling for further statisticalformulations. Note that for a large number of dimensions, the radiuslength distribution corresponds to the sum of numerous independentrandom variables; the central limit theorem states that the resultingdistribution converges to a Laplace-Gauss distribution. In respect withthe Laplace-Gauss modeling limitation, the hypersphere geometry shows ahigher density of measurements on the inside of the shell than on theoutside for a same distance to the shell 203 (as shown in FIG. 3). TheLaplace-Gauss approximation is good when the shell thickness 204 (asshown in FIG. 3) is much smaller than the shell radius 202,corresponding to a large number N of dimensions. The modeling ofLaplace-Gauss dispersion is valid for a large number of dimensions,facilitates implementation of the method and provides an analyticalstatistical predictability of the behavior (e.g., false positive rate,probability of a pattern matching).

In a warning system that monitors a drift magnitude exceeding k times(k>0) the standard deviation σ, knowing that

$\begin{matrix}{{{{{erf}(z)} = {\frac{2}{\sqrt{\; \pi}}{\int_{0}^{z}{e^{- \xi^{2}}\ {\xi}}}}},\; {{{erf}\left( {k/\sqrt{2}} \right)} = {\frac{\sqrt{2}}{\sigma_{n}\sqrt{\pi}}{\int_{0}^{{k\sigma}_{n}}{e^{{{- x^{2}}/2}\sigma_{n}^{2}}\ {x}\mspace{14mu} {and}}}}}}{{{{erf}c}\left( {k/\sqrt{2}} \right)} = {1 - {{erf}\left( {k/\sqrt{2}} \right)}}}} & (12)\end{matrix}$

with the integration of probability density expressed in eq. 11, theprobability to find a measurement inside the shell exceeding the radius202 of the cluster shell “i” by kσ_(i) ^(X−S) is

$\begin{matrix}{\left. {P\left( {X_{m},S_{i}} \right)} \right|_{{({D_{m,i} - r_{i}})} < {k\; \sigma}} = {{0.5 + {\int_{0}^{{+ k}\; \sigma_{i}^{X - S}}{\frac{e^{{{- x^{2}}/2}{(\sigma_{i}^{X - S})}^{2}}}{\sigma_{i}^{X - S}\sqrt{2\; \pi}} \cdot \ {x}}}} = {0.5 + {0.5 \cdot {{{{er}f}\left( {k/\sqrt{2}} \right)}.}}}}} & (13)\end{matrix}$

The probability to find a measurement outside the shell r_(i)+kσ_(i)^(X−S) is

P(X _(m) , S _(i))|_((D) _(m,i) _(−r) _(i)_()>kσ)=0.5−0.5·erf(k/√{square root over (2)}).   (14)

The probability to find a measurement inside the shell r_(i)−kσ_(i)^(X−S) is

P(X _(m) , S _(i))|_((D) _(m,i) _(−r) _(i)_()<−kσ)=0.5−0.5·erf(k/√{square root over (2)}).   (15)

The probability to find a measurement inside and outside the shellboundaries r_(i)±kσ_(i) ^(X−S) are respectively

P(X _(m) , S _(i))|_(|D) _(m,i) _(−r) _(i)_(|<kσ)=1−2·(0.5−0.5·erf(k/√{square root over (2)}))=erf(k/√{square rootover (2)}), and    (16a)

P(X _(m) , S _(i))|_(|D) _(m,i) _(−r) _(i)_(|>kσ)=2·(0.5−0.5·erf(k/√{square root over (2)}))=erfc(k/√{square rootover (2)}).   (16b)

A distance between a measurement 103 and the S_(i) hypersphere shell 203is herein referred to as hypersphere radius deviation (HRD) 210 definedas

HRD_(m,i) =D _(m,i) −r _(i) when {X _(m) :m ∈ [1, M]}  (17a)

and, when a measurement X_(m) does not contribute to the estimatedsignature,

HRD_(m,i) =D _(m,i)−(r _(i)+ε_(i) ^(s)) when {X _(m) :m ∉ [1, M]}  (17b)

For example, in the case of a trending analysis, when the lastmeasurement is compared with a signature average, an alarm may be set byfixing a k maximum value referred to as k_(max). An alarm occurs when

$\begin{matrix}{{k \geq k_{\max}}{with}} & {(18)\mspace{11mu}} \\{k^{2} = {\left( {{HRD}_{m,i}/\sigma_{i}^{X - S}} \right)^{2}.{or}}} & \left( {19a} \right) \\{{k^{2} = {\frac{\left( {D_{m,i} - r_{i}} \right)^{2}}{\left( \sigma_{i}^{X - S} \right)^{2}} = {\frac{\left( {D_{m,i} - r_{i}} \right)^{2}}{r_{i}^{2} \cdot \left( {\frac{1}{N} + \frac{1}{N \cdot \left( {M - 1} \right)}} \right)} = {\left( {\frac{D_{m,i}}{r_{i}} - 1} \right)^{2}/\left( {\frac{1}{N}\left( {1 + \frac{1}{M - 1}} \right)} \right)}}}},{M > 1}} & \left( {19b} \right)\end{matrix}$

when rewritten in terms of measurement distance to estimated signatureD′_(m,i) 205, average measurement distance r_(i) 202, a number M ofmeasurements and a number N of dimensions. The false alarm rate isdefined by eq. 16b with k=k_(max). The method of the invention thus usesthe ratio expressed in eq. 19 with a single statistic parameter k_(max)to monitor the whole measurement pattern.

At start-up, the small M value increases the dispersion (eq. 7 and eq.8). A HRD value observed at start-up appears less significant than thesame deviation observed after collection of more measurements. The kfactor sensitivity increases proportionally to √{square root over(M−1)}, thus reducing the occurrence of false positive at start-up.

When the combined deviation of a plurality of L measurements 103 isconsidered from the signature i, the corresponding k value can beestimated from the average HRD value. Less accurate,

$\begin{matrix}{k^{2} = {{\frac{1}{L}{\sum\limits_{l = 1}^{L}\; \left( \frac{{HRD}_{l,i}}{\sigma_{i}^{X - S}} \right)^{2}}} = {\frac{1}{L}{\sum\limits_{l = 1}^{L}\; k_{l}^{2}}}}} & (20)\end{matrix}$

is another estimation expression for combining many measurements (seeAppendix 1). The same expression can be used for combining a pluralityof L measurement deviations from the same number of correspondingsignatures projected in different

^(N) subspaces when

$\begin{matrix}{k_{l}{\frac{{HRD}_{l,i}}{\sigma_{l}^{X - S}}.}} & (21)\end{matrix}$

Note that in the latter case, since k is dimensionless, the engineeringunit corresponding to the hypersphere subspace l can be fixedarbitrarily. For example, using eq. 20 and eq. 21, it is possible to mixthe deviation of a vibration reading with the deviation observed onmultiple temperature readings.

Referring to FIG. 5, a trend in a signature may be considered assuccessive signatures starting from an initial signature. Differentpossibilities exist for the evolution step between two successivesignatures. A first possibility is easier to detect and corresponds tothe case of (S_(i),S_(i+2)), (S_(i),S_(i+3)) and (S_(i+1),S_(i+3))comparisons. The comparisons (S_(i),S_(i+1)) and (S_(i+1),S¹⁻²) are moredifficult since two hyperspheres are overlapping; some measurements maybe shared by both clusters. The problem is a signature to signaturediscrimination and distance estimation. FIG. 6 shows the worst casewhere the signature step is less than the noise radius and where thenoise fluctuation is also in the same amplitude range as the step. Inthe illustrated case, the measurement dispersion appears larger for thelast signature S_(i+1). These illustrations may be also applied forpattern recognition where a running signature is compared to a signatureextracted from a database. It will be seen that the

^(N) representation of the signal allows discrimination of twosignatures even if the measurement hyperspheres are overlapped.

Referring to FIG. 7, the coordinates of the estimated signature S′_(i)201 in

^(N) appears function of the available corresponding measurements 103(as shown in FIG. 3). Different measurement sets yield differentestimated signature coordinates. There exist an estimation biasE|S_(i)−S′_(i)| and an expectation for the bias dispersion. Asillustrated on the left, two signature estimations S′_(i) 201 and S′_(j)211 are located on their respective signature hyperspheres 300, 301,both centered on their common denoised signature S. As second part ofthe “

^(N) signature statistics” 104 (as shown in FIG. 1), the estimatedsignature S′_(i) hypersphere radius 212

$\begin{matrix}{r_{i}^{s} = {r_{i} \cdot \sqrt{\frac{1}{M_{i}}}}} & (22)\end{matrix}$

decreases when the number of measurements increases. As third part ofthe “

^(N) signature statistics” 104, the signature dispersion 213 (as shownin FIG. 9)

$\begin{matrix}{{\sigma_{i}^{s} = {\sqrt{\frac{1}{M\left( {M - 1} \right)}{\sum\limits_{m = 1}^{M}\; \left( {r_{i} - \sqrt{\sum\limits_{n = 1}^{N}\; \left( {x_{m,n} - S_{i,n}^{\prime}} \right)^{2}}} \right)^{2}}} = {\sigma_{i}^{r}\sqrt{\frac{1}{M - 1}}}}},{M > 1}} & \left( {23\; a} \right)\end{matrix}$

calculated from measurements or estimated

$\begin{matrix}{{\sigma_{i}^{s} = {r_{i} \cdot \sqrt{\frac{1}{2{N\left( {M - 1} \right)}}}}},{M > 1},\left( {{Gaussian}\mspace{14mu} {white}\mspace{14mu} {noise}\mspace{14mu} {assumption}} \right)} & \left( {23b} \right)\end{matrix}$

decreases when the number of dimensions increases. The estimated radiusbias error ε_(i) ^(s) 208 (as shown in FIG. 8) given by eq. 9 is afourth part of the “

^(N) signature statistics” 104 and corresponds to an internal error ofthe signature estimation location in

^(N).

Note that σ_(i) ^(s) has the same mathematical expression as the“dispersion of measurement-to-shell dispersion” σ_(i) ^(r)′ (eq. 8). Insome special cases, these two estimations do not correspond to the samedispersion phenomenon: e.g. considering a merging of two signaturesestimated with different noise amplitudes, the dispersion σ_(i) ^(s)decreases when σ_(i) ^(r)′ may increases with the merging. FIG. 8sums-up the statistic estimations associated with measurements andsignature.

The distance between two hyperspheres

$\begin{matrix}{D_{i,j} = {{{S_{i}^{\prime} - S_{j}^{\prime}}} = \sqrt{\sum\limits_{n = 1}^{N}\; \left( {S_{i,n}^{\prime} - S_{j,n}^{\prime}} \right)^{2}}}} & (24)\end{matrix}$

can be illustrated as one hypersphere centered on a zero-origincoordinate (FIG. 7 right). The signature-to-signature distancehypersphere radius length 214

r _(i,j) ^(s)=√{square root over ((r _(i) ^(s))²+(r _(j) ^(s))²)}{squareroot over ((r _(i) ^(s))²+(r _(j) ^(s))²)}  (25a)

is the quadratic sum of the respective hypersphere radius lengths.Including the internal error in the radius length, the signaturedistance hypersphere radius length 214 can be rewritten

r _(i,j) ^(sε)=√{square root over ((r _(i,j) ^(s))²+(ε_(i) ^(s))²+(ε_(j)^(s))² )}{square root over ((r _(i,j) ^(s))²+(ε_(i) ^(s))²+(ε_(j) ^(s))²)}{square root over ((r _(i,j) ^(s))²+(ε_(i) ^(s))²+(ε_(j) ^(s))² )}.  (25b)

A distance between two signatures is herein referred to as hyperspheresignature deviation (HSD) 215 defined as

HSD_(i,j) =D _(i,j)−√{square root over ((r _(i,j) ^(s))²+(ε_(i)^(s))²+(ε_(j) ^(s))² )}{square root over ((r _(i,j) ^(s))²+(ε_(i)^(s))²+(ε_(j) ^(s))² )}{square root over ((r _(i,j) ^(s))²+(ε_(i)^(s))²+(ε_(j) ^(s))² )}=D _(i,j) −r _(i,j) ^(sε)  (26)

(see Appendix 2).

FIG. 9 illustrates a signature-to-signature comparison. The estimatedsignature S′_(i) hypersphere radius 212 given at eq. 22 is used in thecase of uniform average. For a moving average signature with non-uniformweighting, the radius 212 exists with a different mathematicalexpression. Whatever the mathematical expression of averaging and thecorresponding dispersion, the total dispersion

(σ_(i,j) ^(S−S))²=(σ_(i) ^(s))²+(σ_(j) ^(s))²   (27)

between signatures 216 is related to HSD 215 in order to obtain theprobability density

$\begin{matrix}{{\Omega \left( {S_{i,}S_{j}} \right)} = \frac{^{{{- {HSD}_{i,j}^{2}}/2}{(\sigma_{i,j}^{S - S})}^{2}}}{\left( {\sigma_{i,j}^{S - S}\sqrt{2\; \pi}} \right.}} & (28)\end{matrix}$

in the case of a Laplace-Gauss dispersion modeling. Note that whenσ_(i,j) ^(S−S)<<r_(i,j) ^(s), the

^(N) representation of the signal may allow discrimination of twosignatures even if the signature hyperspheres overlap. From eq. 12 andwith integration of probability density expressed in eq. 28, theprobability to find the moving average signature inside the k·σ_(i,j)^(S−S) limit is

$\begin{matrix}{\left. {P\left( {S_{i},S_{j}} \right)} \right|_{D_{i,j} < {k\; \sigma}} = {{\int_{0}^{{+ k}\; \sigma_{i,j}^{S - S}}{\frac{^{{{- {HSD}_{i,j}^{2}}/2}{(\sigma_{i,j}^{S - S})}^{2}}}{\sigma_{i,j}^{S - S}\sqrt{2\; \pi}}\  \cdot {x}}}\; = {{{erf}\left( {k/\sqrt{2}} \right)}.}}} & (29)\end{matrix}$

For example, in a signature-to-signature comparison, an alarm may be setby fixing a k maximum value k_(max). An alarm occurs when k≧k_(max) with

k=HSD_(i,j)/σ_(i,j) ^(S−S).   (30)

The false alarm rate is defined by eq. 16b with k=k_(max). The method ofthe invention thus uses the ratio expressed in eq. 30 with a singlestatistic parameter k_(max) to monitor the whole signature patternmatching Similarly to the merging process proposed at eq. 20 and eq. 21,the expression

$\begin{matrix}{k^{2} = {{\frac{1}{L}{\sum\limits_{l = 1}^{L}\; \left( \frac{{HSD}_{l}}{\sigma_{l}^{S - S}} \right)^{2}}} = {\frac{1}{L}{\sum\limits_{l = 1}^{L}\; k_{l}^{2}}}}} & (31)\end{matrix}$

may be used to merge the probability of L signature-to-signaturecomparisons.

The method of the invention is optimal when noise amplitude is uniformlydistributed among the measurements such as set forth in eq. 1. Whennoise amplitude is non-uniformly distributed, a normalization

$\begin{matrix}{{X_{mn}^{\prime} = {X_{mn} \cdot \frac{1}{\sigma_{n}}}},{n \in \left\lbrack {1,N} \right\rbrack}} & (32)\end{matrix}$

of measurements with

$\begin{matrix}{{{\sigma_{n}^{2} = {\frac{1}{M - 1} \cdot {\sum\limits_{m = 1}^{M}\; \left( {X_{m,n} - {\overset{\_}{X}}_{n}} \right)^{2}}}},{M > 1}}{and}} & (33) \\{{\overset{\_}{X}}_{n} = {\frac{1}{M} \cdot {\sum\limits_{m = 1}^{M}\; X_{m,n}}}} & (34)\end{matrix}$

yields a uniform standard deviation through the different measurements.A drawback of such a normalization may occur when a measuring problem(e.g. amplitude clipping) or another unexpected event decreases thedispersion of some measurement samples; with the proposed normalization,more weight is assigned to such biased measurement samples. Thenormalization

$\begin{matrix}{{X_{mn}^{\prime} = {X_{mn} \cdot \sqrt{\frac{2}{{\overset{\_}{\sigma}}^{2} + \sigma_{n}^{2}}}}},{n \in {\left\lbrack {1,N} \right\rbrack {with}}}} & \left( {35a} \right) \\{\overset{\_}{\sigma} = {\frac{1}{N} \cdot {\sum\limits_{n = 1}^{N}\; \sigma_{n}}}} & \left( {35b} \right)\end{matrix}$

is an example of normalization which may partially overcome thedrawback.

In some cases, the noise appears function of the measured signalamplitude. WO 2012/162825 (Léonard) proposes the use of a dedicatedmetric to distort the subspace in order to obtain a cluster close to ahypersphere.

Referring to FIG. 10, when a measurement may be a member of onesignature among many possible signatures, the

^(N) visualisation is reversed between measurement and signature toresolve the classification problem. As illustrated, the measurementhypersphere 400 is centered with the measurement. The measurementhypersphere radius 401

r=√{square root over (NE(n _(mn) ²))}  (36)

corresponds to the average noise vector length projected in

^(N). When only one or a few measurements are available to feed eq. 6,it will be more accurate to replace eq. 36 by a local average ofsignature hypersphere radius in the vicinity of the measurement,assuming that the noise amplitude is similar for the surroundingsignatures. In the illustrated case of similar signature dispersionσ_(i) ^(s) for S′₁ to S′₅, the measurement maximum membership likelihoodis obtained for the signature S′₄.

In this context, the hypersphere radius deviation (HRD 211) is thedistance between the signature and the measurement hypersphere 400. Thedensity probability function

$\begin{matrix}{{\Omega \left( {X_{m},S_{i}} \right)} = \frac{e^{{{- {({D_{m,i} - r_{i}})}^{2}}/2}{(\sigma_{i}^{X - S})}^{2}}}{\sigma_{i}^{X - S}\sqrt{2\; \pi}}} & (37)\end{matrix}$

appears similar to that expressed at eq. 11 in the case of aLaplace-Gauss dispersion modeling. The membership likelihood of themeasurement “m” for the signature “i”

$\begin{matrix}{{P\left( {X_{m} \in S_{i}} \right)} = \frac{\Omega \left( {X_{m},S_{i}} \right)}{\sum\limits_{j}\; {\Omega \left( {X_{m},S_{j}} \right)}}} & (38)\end{matrix}$

is related to the sum of all density probability functions existing inthe vicinity of the measurement. Note that P(X_(m)∈S_(i))=1 when onlyone signature is considered (e.g. in trending).

Referring to FIG. 11, looking at the

^(N) projection of many measurement sets, the estimated signatureS′_(i,x) 201 associated with one of these sets, with the correspondingmeasurement hypersphere radius 202, appears in the shell of anhypersphere 300 having its center close to the real signature S_(i) 100.When the other signature estimations are built with the same measurementpopulation and similar noise amplitude, the signature locations areclose to a common signature hypersphere shell 300. In other words,similar signature estimations appear located in the shell of a smallerhypersphere centered on the real signature S_(i).

The signatures may be processed like individual measurements 103 with anestimated signature, a hypersphere radius and hardness. The measurementset may be split into many signatures by a simple uniform averaging overm_(a) measurements such as

$\begin{matrix}{X_{m_{a},l} = {\frac{1}{m_{a}}{\sum\limits_{m = 1}^{m_{a}}\; {X_{m + {1 \cdot m_{a}}}.}}}} & (39)\end{matrix}$

The initial measurement sets of M realizations became a signature setshowing a population of M/m_(a). The estimated signature from M/m_(a)signatures is the same as the signature obtained from M measurements,but the

^(N) measurement statistics 105 (as shown in FIG. 1) differ. Thetrending and pattern matching described earlier for a set ofmeasurements is also valid for the set of signatures. The action ofgenerating a signature from other signatures or from initialmeasurements is herein referred to as averaging step. An evolution stepmay for example correspond to an averaging step of successivemeasurements or successive evolution steps. In other words, the proposedmethod may include some recursive aspects.

The respective hypersphere radius, measurement dispersion and estimationerror on measurement dispersion for an initial measurement set of Mrealizations can be expressed as function of noise energy expectation η₀²=E(n_(mn) ²) as

$\begin{matrix}{{r_{i} = {\eta_{0}\sqrt{N}}},{\sigma_{i}^{r} = {{\eta_{0}\sqrt{1/2}\mspace{14mu} {and}\mspace{14mu} \sigma_{i}^{\prime \; r}} = {\eta_{0} \cdot \sqrt{\frac{1}{2 \cdot \left( {M - 1} \right)}}}}}} & (40)\end{matrix}$

from eq. 6, eq. 7 and eq. 8. Splitting the measurement set in M/m_(a)averaged measurements reduces the noise energy expectation of theresulting signatures by a m_(a) factor. The averaged measurementhypersphere radius and shell dispersion

$\begin{matrix}{r_{m_{a},l} = {{{\eta_{0} \cdot \sqrt{N \cdot \frac{1}{m_{a}}}}\mspace{14mu} {and}\mspace{14mu} \sigma_{m_{a},i}^{r}} = {\eta_{0} \cdot \sqrt{\frac{1}{2\; m_{a}}}}}} & (41)\end{matrix}$

appear stretched compared to the original set of M measurements when theestimation error on measurement dispersion

$\begin{matrix}{{\sigma_{m_{a},i}^{\prime\gamma} = {{{\eta_{0} \cdot \sqrt{\frac{1}{2{m_{a} \cdot \left( {\frac{M}{m_{a}} - 1} \right)}}}}\mspace{14mu} {or}} \approx {{\eta_{0} \cdot \sqrt{\frac{1}{2\left( {M - 1} \right)}}}\mspace{14mu} {for}\mspace{14mu} M}}}\operatorname{>>}m_{a}} & (42)\end{matrix}$

appears approximately unchanged. An interesting fact is that the ratior/σ^(r) appears unchanged through the splitting transform when the ratior/σ′^(r) decreases: the

_(M) ^(N)→

_(M/m) _(a) ^(N) averaging step stretches unequally the hyperspheregeometric characteristics. By inserting eq. 40 into eq. 10, the totaldispersion expressed in terms of noise expectation energy is

$\begin{matrix}{\left( \sigma_{m_{a},i}^{X - S} \right)^{2} = {{\left( \sigma_{i}^{r} \right)^{2} + \left( \sigma_{i}^{\prime \; r} \right)^{2}} = {{{0.5 \cdot \eta_{0}^{2}} + \frac{\eta_{0}^{2}}{2 \cdot \left( {M - 1} \right)}} = {\frac{1}{2}{\left( \frac{M}{M - 1} \right) \cdot {\eta_{0}^{2}.}}}}}} & (43)\end{matrix}$

The corresponding total dispersion after the

_(M) ^(N)→

_(M/m) _(a) ^(N) averaging step is

$\begin{matrix}{{\left( \sigma_{m_{a},i}^{X - S} \right)^{2} = {{\frac{1}{2\; m_{a}}\eta_{0}^{2}} + {\frac{1}{2\; {m_{a} \cdot \left( {\frac{M}{m_{a}} - 1} \right)}} \cdot \eta_{0}^{2}}}}{or}} & (44) \\\left. \left( \sigma_{m_{a},i}^{X - S} \right)^{2} \middle| {}_{M\operatorname{>>}m_{a}}{\approx {\frac{1}{2}{\left( {\frac{1}{m_{a}} + \frac{1}{M}} \right) \cdot {\eta_{0}^{2}.}}}} \right. & (45)\end{matrix}$

The

_(M) ^(N)→

_(M/m) _(a) ^(N) averaging step significantly reduces the totaldispersion with the drawback of an increased response time of m_(a)sample delay.

The average of m_(a) measurements may be compared to a signature using

D _(l,i) =∥X _(m) _(a) _(,l) −S′ _(i)∥=√{square root over (Σ_(n=1)^(N)(X _(m) _(a) _(,l,n) −S′ _(i,n))²)}  (46)

from eq. 5. In that case, the

^(N) measurement statistics 105 (as shown in FIG. 1) of the averagemeasurements may be estimated from eq. 7 and eq. 8 using X_(m) _(a)_(,l) instead X_(m), or rewritten as function of the statistics obtainedwithout measurement averaging such as

$\begin{matrix}{{r_{m_{a}i} = {r_{i} \cdot \sqrt{\frac{1}{m_{a}}}}},} & (47) \\{{\sigma_{m_{a},i}^{r} = {\sigma_{i}^{r} \cdot \sqrt{\frac{1}{m_{a}}}}}{and}} & (48) \\{\sigma_{m_{a},i}^{\prime \; r} = {\sigma_{i}^{\prime \; r} \cdot {\sqrt{\frac{\left( {M - 1} \right)}{\left( {M - m_{a}} \right)}}.}}} & (49)\end{matrix}$

The signature location error

ε_(m) _(a) _(,i) ^(s)=ε_(i) ^(s)   (50)

is unchanged since the error is only function of the M measurement setused for signature estimation.

The HRD of averaged measurements is

HRD_(l,i) =D _(l,i) −r _(m) _(a) _(,i) when {X _(m+l·m) _(a) :m+l·m _(a)∈ [1, M]} in eq. 39   (51)

and, when the measurement X_(m) does not contribute to the estimatedsignature

HRD_(l,i) =D _(l,i)−(r _(m) _(a) _(,i)+ε_(i) ^(s)) when {X _(m+l·m) _(a):m+l·m _(a) ∉ [1, M]} in eq. 39   (52)

Referring to FIG. 12, there is shown a graph illustrating concatenatedtime series of histograms generated using 250 vibroacoustic measurementstaken from an electrical equipment in operation. The horizontal axisrepresents time expressed in the form of sample numbers of themeasurements, the vertical axis represents the amplitude of themeasurements in dB, and the measurements have tones according to theshading legend as function of sample count. Processing of themeasurements according to the method of the invention produces a

^(N) hypersphere similar to that shown in FIG. 3 and a signature driftsimilar to that shown in FIG. 5 due to a drift between the samples450-500. FIG. 13 shows a curve 300 representing the radius deviationcomputed on the vibroacoustic measurements of FIG. 12 according to themethod of the invention, and alarm settings 301, 302 fixed respectivelyto +4σ and −4σ. In the illustrated example, the hypersphere radiusreaches a value of 34.2171 with a hypersphere hardness of 1.14184 and asignature dispersion of 4.88816. To sum up, the projection of theN-samples signal in a multidimensional space

^(N) for several realizations of measurement shows a distribution closeto a hypersphere of radius r, where r corresponds to the statisticalaverage of measurement dispersion. At the center of the hypersphere islocated the estimated signature S′_(i). The statistical “thickness” ofthe hypersphere shell is the “hardness” of the hypersphere. As a resultof the “sphere hardening phenomenon”, the hardness to radius ratiodecreases with the increasing number N of dimensions. The measurementprobability density appears as function of the distance between themeasurement and the hypersphere surface, with the distance related tothe hardness. In the case of a large number of dimensions, themeasurement distance to the hypersphere shell density probabilityfunction converges to a Laplace-Gauss modeling (central limit theorem)and yields an analytical statistical formulation for the measurementlikelihood. The proposed method gradually increases the sensitivity withthe new measurements. Occurrence probability of “false positive” (falsealarm) appears constant from start-up through steady state: the falsealarm rate at start-up is similar to that occurring in steady state.

Referring to FIG. 14, let's consider the N samples measurementX_(m)=(X_(1,m), . . . , X_(i,m), . . . , X_(N,m))^(T) expressed in eq. 1as a m-th realization of a vector of random variables X_(n), each with afinite variance. A set of M realisations is used in the estimation ofthe covariance matrix

Σ_(i,j)=cov(X _(i,m) , X _(j,m))=E[(X _(i,m)−μ_(i))·(X_(j,m)−μ_(j))]  (53)

where

μ_(i)=E[X_(i,m)]  (54)

is the expected value of the i-th entry in the vector X. Expressed as

$\begin{matrix}{\sum{= {\quad{\begin{pmatrix}{E\left\lbrack {\left( {X_{1,m} - \mu_{1}} \right) \cdot \left( {X_{1,m} - \mu_{1}} \right)} \right\rbrack} & K & {E\left\lbrack {\left( {X_{1,m} - \mu_{1}} \right) \cdot \left( {X_{N,m} - \mu_{N}} \right)} \right\rbrack} \\M & O & M \\{E\left\lbrack {\left( {X_{N,m} - \mu_{N}} \right) \cdot \left( {X_{1,m} - \mu_{1}} \right)} \right\rbrack} & L & {E\left\lbrack {\left( {X_{N,m} - \mu_{N}} \right) \cdot \left( {X_{N,m} - \mu_{N}} \right)} \right\rbrack}\end{pmatrix},}}}} & (55)\end{matrix}$

the matrix appears square and symmetric. The same logic used for the

^(N) projection of a signature is now applied for the projection of thecovariance matrix in

^(NN). The N-dimensionality of a signature is replaced by theN×N-dimensionality of a variance.

Projected in the subspace

^(NN), the M cross-measurement matrices

$\sum_{i,j,m}{= \left\lbrack {\left( {X_{i,m} - {\overset{\_}{X}}_{i}} \right) \cdot \left( {X_{j,m} - {\overset{\_}{X}}_{j}} \right)} \right\rbrack}$with${\overset{\_}{X}}_{i} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}X_{i,m}}}$

generate a cluster of points, each one corresponding to a measurement.The estimated covariance matrix

$\begin{matrix}{\sum_{i,j}^{\prime}{= {{\frac{1}{M}{\sum\limits_{m = 1}^{M}\sum_{i,j,m}}} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}\left\lbrack {\left( {X_{i,m} - {\overset{\_}{X}}_{i}} \right) \cdot \left( {X_{j,m} - {\overset{\_}{X}}_{j}} \right)} \right\rbrack}}}}} & (56)\end{matrix}$

corresponds to a point located at the mass center of thecross-measurement matrix cluster.

As for the signature trending, the updated estimated covariance matrixand the following other estimations may be generated by a moving averageprocess (uniform average is presented here to keep the mathematicalexpressions short). In the case of an Euclidian metric in

^(NN) with uniform averaging, the cross-measurement matrices toestimated covariance matrix distance

$\begin{matrix}{D_{m,i} = {{{\sum_{i,j,m}{- \sum_{i,j}^{\prime}}}} = \left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\left( {{\left( {X_{i,m} - {\overset{\_}{X}}_{i}} \right)\left( {X_{j,m} - {\overset{\_}{X}}_{j}} \right)} - \sum_{i,j}^{\prime}} \right)^{2}}} \right)^{\frac{1}{2}}}} & (57)\end{matrix}$

has the expected mean distance

$\begin{matrix}{{r_{i} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}{D_{m,i}\mspace{14mu} {for}\mspace{14mu} X_{m}}}} \in {{{cluster}\mspace{14mu}}^{``}i^{''}}}},} & \left( {58a} \right)\end{matrix}$

or (possibly less accurate)

$\begin{matrix}{{r_{i} = {{\sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}D_{m,i}^{2}}}\mspace{14mu} {for}\mspace{14mu} X_{m}} \in {{{cluster}\mspace{14mu}}^{``}i^{''}}}},} & \left( {58b} \right)\end{matrix}$

for a given measurement set. This estimation is referred to ascovariance hypersphere radius and the double of the expected standarddeviation

$\begin{matrix}{\sigma_{i}^{r} = \sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}\left( {D_{m,i} - r_{i}} \right)^{2}}}} & \left( {59a} \right)\end{matrix}$

is referred to as thickness of the cross-measurement matrix distance toradius distribution calculated from the cross-measurement matrices orestimated

$\begin{matrix}{\sigma_{i}^{r} = {{r_{i} \cdot \sqrt{\frac{1}{2N}}}\mspace{14mu} \left( {{Gaussian}\mspace{14mu} {white}\mspace{14mu} {noise}\mspace{14mu} {assumption}} \right)}} & \left( {59b} \right)\end{matrix}$

from the estimated covariance hypersphere radius length. In the case ofGaussian white noise, the estimated dispersion (eq. 59b) appears withless dispersion than the calculated one (eq. 59a). For a small M set,the estimation error on the standard deviation, the dispersion ofmeasurement dispersion

$\begin{matrix}{\mspace{79mu} {{\left( \sigma_{i}^{\prime \; r} \right)^{2} = {\frac{\sigma_{i}^{r\; 2}}{M - 1}\mspace{14mu} {or}}}\mspace{14mu} {\sigma_{i}^{r^{\prime}} = {{r_{i} \cdot \sqrt{\frac{1}{2{N \cdot \left( {M - 1} \right)}}}}\mspace{14mu} \left( {{Gaussian}\mspace{14mu} {white}\mspace{14mu} {noise}\mspace{14mu} {assumption}} \right)}}}} & (60)\end{matrix}$

should be also taken in account. Finally, the estimated radius biaserror

$\begin{matrix}{{ɛ_{i}^{s} = {r_{i} \cdot \sqrt{\frac{1}{M\left( {M - 1} \right)}}}},{M > 1},} & (61)\end{matrix}$

corresponds to an additional radius length observed for a measurementwhich is not included in the computed estimated covariance: {X_(m):m ∉[1, M]}.

The further development of covariance trending and pattern recognitionis similar to that developed for signatures.

FIG. 14 illustrates the measurements shown in FIG. 12 transposed in theestimated covariance.

In multidimensional space

^(N), a signature for a given operating conditions appears as a point.If the signature changes with one of the operating conditions, e.g. withthe temperature, the signature domain 500 for a temperature rangecorresponds to a path in

^(N) as shown in FIG. 17. In the illustrated example, the path is movingfaster at lower temperatures. The measurement standard deviation canalso change with the operating conditions. FIG. 18 shows the dispersionof measurements around the signature domain 500. For a point on the pathshown in FIG. 17 corresponding to a given temperature, eachcorresponding realization appears in the shell of a hypersphere centeredto this point. If a superposition of hypersphere shells centered on thesignature domain 500 is considered, the resulting density function is acylindrical shell centered with respect to the signature domain 500. Inthe case of measurement standard deviation function of the temperature,the cylindrical radius appears variable and as function of the standarddeviation as illustrated in FIG. 18.

If the signature changes with two of the operating conditions, e.g. withthe temperature and the load, the signature domain then corresponds to a2-D surface in

^(N). The density probability function of a realisation in

^(N) appears like a shell enveloping the signature surface domain. Thestandard deviation of the realizations sets the distance between theshell and the signature domain.

In the current context, the measurement domain has N dimensions, thesignature domain has L dimensions, and the number of descriptivevariables are the minimum number of variables needed to describe adomain. For example, a hypersphere in

^(N) has N-1 dimensions. Also called a n-sphere with n=N−1 in that case,the location of a point onto its surface can be achieved with N−1coordinates but the mathematical description of the n-sphere needs n+2descriptive variables (i.e. N coordinates plus the radius). The casewhere a signature changes with one of the operating conditionscorresponds to one dimension but the cylindrical shell, representativeof the center of the density probability function of a realisation,enveloping the signature domain has N−1 dimensions.

Referring to FIG. 19, at this point we will generalize the

^(N) statistics to

^(N,L). The one point domain signature associated to a hyperspherecorresponds to a

^(N,0), with L=0 dimension for a point. The 1D path of a signaturesensitive to one of the operating conditions corresponds to the

^(N,1) case. The 2D signature sensitive to two of the operatingconditions corresponds to the

^(N,2) case and so on. If the signature changes with L of the operatingconditions, the signature domain for an operating condition rangecorresponds to an N,L-manifold 501 in

^(N). The density probability function of a realisation in

^(N) appears like a manifold enveloping the N,L-manifold signaturedomain 501.

A manifold is a topological space showing the same properties of theEuclidean space near each point element of the manifold. Locally, closeto the manifold, the Euclidian distance can be used. However, on agreater scale, the manifold shape adds a bias to a Euclidian distanceestimator and globally, a manifold might not resembles a Euclideanspace. The 1D path of a signature is an open manifold and also the 2Dsignature in

^(N). A hypersphere is a closed manifold and, for the general case

^(N,L), the enveloping manifold associated to the density probabilityfunction of a realisation is also a closed manifold.

A new geometric figure called “shell manifold” is herein introduced. Ashell manifold is a manifold enveloping a second manifold and has atleast one additional dimension called thickness. Locally, the thickness506 is oriented perpendicular to the signature manifold 501. In thecontext of the proposed method, the shell manifold defines the densityprobability function of a realisation in

^(N). In this context, the shell thickness 506 is defined as the doubleof the standard deviation of the measurement distance 510 with respectto the signature manifold 501. The shell “center” is representative ofthe mass center of the density probability function of a realisation.The Laplace-Gauss density probability function has its maximum ofdensity in coincidence with the shell center. A shell center and athickness are enough to represent a Laplace-Gauss density probabilityfunction but it is not the case for Poisson law and many otherdistribution laws having their mass center not aligned with theirmaximum and requiring some additional parameters to describe theirdistribution. In practice, with numerous dimensions N, the central limittheorem can be applied locally and a section of the shell manifold canbe described by the Laplace-Gauss function. Moreover, in the case ofnon-constant distance between the enveloping manifold and the signature(i.e. the measurement standard deviation appears function of thecoordinates which define a point into a signature domain), comparativelyto a N,L-manifold associated to a signature, many additional descriptivevariables are required to define the enveloping shell manifold shape ordensity. In the simplest case of constant amplitude Laplace-Gauss noiseaddition over the N dimensions, the distance shell to signature manifoldand the shell thickness are constant. It should be noted that takenlocally in the vicinity of the shell, the density probability functionof a realisation fills the

^(N) volume if the function is not bound (e.g. Laplace-Gauss function):the shell manifold corresponding to the density probability function ofa realisation has N+1 dimensions (N coordinates plus the density).

The signature manifold may be built from the averaging of the availablerealizations. While a simple realization addition (eq. 4) works for thecase L=0, the averaging method appears more complex for L>0 since a fitmust be made to model the manifold pattern. In that case, the kriggingmethod (see e.g. dual krigging method developed by Matheron) may be thebest unbiased linear interpolator and takes account of the localdispersion of the measurements. Before krigging, the merging of closerealizations may allow a first estimation of the local dispersion andreduce the computing task.

Locally, the topological space of the shell manifold shows the sameproperties of the Euclidean space on the manifold. Perpendicularly tothe shell manifold, the same Euclidian space properties also exist for asmall distance to the manifold, when the measurement distance to shellcenter 512 is smaller than local manifold curvature. For thoseconditions, the statistic rules to be used for the general case of

^(N,L) appear similar to the equation set defined for a hypersphere with

^(N,0). However, the radius (shell-to-signature domain average distance)r 504 must be corrected to take account of the part 514 of themeasurement noise projected onto the signature domain 501. In thesimplest case of realization noise described by a Laplace-Gaussdistribution centered with the same amplitude for all dimensions, theradius length correction can be written as

$\begin{matrix}{r_{L} = {r_{L = 0} \times \sqrt{\frac{N - L}{N}}}} & (62)\end{matrix}$

where r_(L=0) corresponds to the r_(i) (eq. 6) for a given operatingcondition set. In that case, the distance r 504=r_(L) corresponds to thedistance between the shell center 508 and the signature manifold 501 asillustrated in FIG. 19.

An aspect of the method of the invention is that it maps the measurementdomain like a shell manifold enveloping a signature manifold. In thisvision, the measurements are elements of a hollow cloud whereas inclassical statistics, in contrast, measurements appear inside the cloud.In classical statistics, the characteristic distances of the measurementdensity probability function are the distances of the measurements withrespect to a signature manifold. In the proposed method, thecharacteristic distances of the measurement density probability function516 are the distances 512 of the signal related measurements withrespect to a shell manifold 502 located around a signature manifold 501.In the case of numerous dimensions N, a Laplace-Gauss function can beused and the distances 512 of the signal related measurements 103 arewith respect to a shell manifold center 508.

Hollow clouds of realizations involve an adaptation of severalmathematical tools. For example, the Gaussian mixture model (GMM) is atool to be adapted to hollow clouds. Used in various algorithms, thekernel trick is also subjected to an adaptation. Indeed, the Gaussiankernel

$\begin{matrix}{{K\left( {x,y} \right)} = {\exp\left( {- \frac{{{x - y}}^{2}}{2\sigma^{2}}} \right)}} & (63)\end{matrix}$

between two input vectors x and y refers to a centered Laplace-Gaussdispersion N(0, σ²). With the method of the invention, the averagedistance between two realizations x and y in

^(N) has a √{square root over (r_(x) ²+r_(y) ²)} magnitude, that is theroot mean square of the magnitudes of the noise vectors in

^(N). The magnitude of the average noise vector r may be given byequation 6a when there is a signature mapped to the input vector orestimated otherwise. With the method of the invention, the kernel

$\begin{matrix}{{K\left( {x,y} \right)} = {\exp\left( {- \frac{\left( {{{x - y}} - \left( {r_{x}^{2} + r_{y}^{2}} \right)^{1/2}} \right)^{2}}{2\left( {\sigma_{rx}^{2} + \sigma_{ry}^{2}} \right)}} \right)}} & (64)\end{matrix}$

corresponds to a non-centered Laplace-Gauss dispersion N(√{square rootover (r_(x) ²+r_(y) ²)}, √{square root over (σ_(rx) ²+σ_(ry) ²)}) whereσ_(rx) and σ_(ry) correspond to the dispersion of the noise vectorsr_(x) and r_(y). Likewise, as a first approximation is the kernel

$\begin{matrix}{{K\left( {x,y} \right)} = {\exp\left( {- \frac{\left( {{{x - y}} - r_{x}} \right)^{2}}{2\sigma_{rx}^{2}}} \right)}} & (65)\end{matrix}$

for the comparison between a realization x and a signature y. A moreaccurate value

$\begin{matrix}{{K\left( {x,y} \right)} = {\exp\left( {- \frac{k^{2}}{2}} \right)}} & (66)\end{matrix}$

is obtained from the k² factor given at equation 19a. If the k² factorgiven at equation 30 is inserted into the previous equation, the kernelfor x and y inputs corresponding to signatures (or local averages ofrealizations) is obtained.

The kernel trick is used in statistics and in machine learning. In theprincipal component analysis (PCA), the covariance matrix is impacted bythe method of the invention as well as the kernel trick. The supportvector machines (SVMs) frequently use the Gaussian kernel to be adaptedfor hollow clouds as described above. The hollow cloud concept doublyimpacts the linear discriminant analysis (LDA): the multivariate normaldistribution and the Gaussian kernel used are affected.

While embodiments of the invention have been illustrated in theaccompanying drawings and described above, it will be evident to thoseskilled in the art that modifications may be made therein withoutdeparting from the invention.

Appendix 1—Merging Dimensions

Given a set of M measurements of N dimensions separated in twomeasurement sets comprising respectively N₁ and N₂ dimensions withN=N₁+N₂, since the dimensions share a same engineering unit, they can bemixed and shared.

The square distance of the sets can be added

$\begin{matrix}{D_{m,i}^{2} = {{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}} = {{{\sum\limits_{n = 1}^{N_{1}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}} + {\sum\limits_{n = {1 + N_{1}}}^{N_{1} + N_{2}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}} = {D_{m,i,1}^{2} + D_{m,i,2}^{2}}}}} & \left( {{A1}\text{-}1} \right)\end{matrix}$

The square radius of the sets can be added

$\begin{matrix}{r_{i}^{2} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}}} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}\left( {{\sum\limits_{n = 1}^{N_{1}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}} + {\sum\limits_{n = {1 + N_{1}}}^{N_{1} + N_{2}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}} \right)}}}} & \left( {{A1}\text{-}2} \right) \\{= {{{\frac{1}{M}{\sum\limits_{m = 1}^{M}\left( {\sum\limits_{n = 1}^{N_{1}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}} \right)}} + {\frac{1}{M}{\sum\limits_{m = 1}^{M}\left( {\sum\limits_{n = {1 + N_{1}}}^{N_{1} + N_{2}}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}} \right)}}} = {r_{i,1}^{2} + r_{i,2}^{2}}}} & \left( {{A1}\text{-}3} \right)\end{matrix}$

The square addition is also applicable to signature dispersion

$\begin{matrix}{\left( \sigma_{i}^{s} \right)^{2} = {{\frac{1}{M - 1} \cdot r_{i}^{2}} = {{\frac{1}{M - 1} \cdot \left( {r_{i,1}^{2} + r_{i,2}^{2}} \right)} = {\left( \sigma_{i,1}^{s} \right)^{2} + \left( \sigma_{i,2}^{s} \right)^{2}}}}} & \left( {{A1}\text{-}4} \right)\end{matrix}$

But the square addition is not true for hypersphere shell dispersion(hardness)

$\begin{matrix}{\left( \sigma_{i}^{r} \right)^{2} = {{\frac{1}{M}{\sum\limits_{m = 1}^{M}{\frac{1}{N}{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}}}} = {{{\frac{1}{N} \cdot \frac{1}{M}}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i,n}^{\prime}} \right)^{2}}}} = {\frac{1}{N} \cdot r_{i}^{2}}}}} & \left( {{A1}\text{-}5} \right) \\{= {{\frac{1}{N} \cdot \left( {r_{i,1}^{2} + r_{i,2}^{2}} \right)} = {{\frac{1}{N} \cdot \left( {{N_{1}\left( \sigma_{i,1}^{r} \right)}^{2} + {N_{2}\left( \sigma_{i,2}^{r} \right)}^{2}} \right)} = {{\frac{N_{1}}{N} \cdot \left( \sigma_{i,1}^{r} \right)^{2}} + {\frac{N_{2}}{N} \cdot \left( \sigma_{i,2}^{r} \right)^{2}}}}}} & \left( {{A2}\text{-}6} \right)\end{matrix}$

and is also not true for square deviation

(D _(m,i) −r)²≦(D _(m,i,1) r ₁)²+(D _(m,i,2) r ₂)²   (A1-7)

Comparing

$\begin{matrix}{{k^{2} = {\frac{\left( {D_{m,i} - r_{i}} \right)^{2}}{\left( \sigma_{i}^{X - S} \right)^{2}} = {\left( {\frac{D_{m,i}}{r_{i}} - 1} \right)^{2}/\left( {\frac{1}{N\left( {M - 1} \right)} + \frac{1}{N}} \right)}}}{with}} & \left( {{A1}\text{-}8} \right) \\{k^{2} = {{\frac{1}{L}{\sum\limits_{l = 1}^{L}\left( \frac{{HMD}_{l}}{\sigma_{i}^{X - S}} \right)^{2}}} = {\frac{1}{L}{\sum\limits_{l = 1}^{L}k_{l}^{2}}}}} & \left( {{A1}\text{-}9} \right)\end{matrix}$

the k value obtained for merged measurements with eq. A1-8 is not equalto the proposed estimation with eq. A1-9: both numerator (see inequalityA1-7) and denominator (if N₁≠N₂) work for the divergence. These valuesare close when the number of dimensions is similar and when thedeviation to real signature is uniformly distributed between samples.

In respect with signature-to-signature comparison, eq. A1-1 demonstratesthat the eq. 28 used to combine the k factor for signatures separatedinto two sets

$\begin{matrix}{k^{2} = {\frac{D_{m,i}^{2}}{\left( \sigma_{i}^{s} \right)^{2} + \left( \sigma_{j}^{s} \right)^{2}} \approx {\frac{D_{m,i,1}^{2}}{\left( \sigma_{i,1}^{s} \right)^{2} + \left( \sigma_{j,2}^{s} \right)^{2}} + \frac{D_{m,i,2}^{2}}{\left( \sigma_{i,1}^{s} \right)^{2} + \left( \sigma_{j,2}^{s} \right)^{2}}}}} & \left( {{A1}\text{-}10} \right)\end{matrix}$

can yield an accurate result when the separated signatures show similarstandard deviation. From eq. 9

$\sigma_{i}^{s} = {r_{i} \cdot \sqrt{\frac{1}{M - 1}}}$

the signature dispersion appears similar if the hypersphere radius aresimilar. Since hypersphere radius is noise by the square root of thenumber of dimensions N, the merging of the k factor using eq. 30 forpieces of signatures having similar dimensionality appears accurate.

In conclusion, eq. 20 and eq. 30 are not exact formulations to combine kvalues. They are useful approximations showing an error increasing withthe discrepancies between the number of merged dimensions. They areespecially useful to combine measurement sets having differentengineering units.

Appendix 2—Internal Error Included in Signature Radius

Including the internal error in the signature radius as

$\begin{matrix}{\left( r_{i}^{S\; ɛ} \right)^{2} = {{\left( r_{i}^{S} \right)^{2} + \left( ɛ_{i}^{s} \right)^{2}} = {{{r_{i}^{2} \cdot \frac{1}{M}} + {r_{i}^{2} \cdot \frac{1}{M\left( {M - 1} \right)}}} = {r_{i}^{2} \cdot \frac{1}{\left( {M - 1} \right)}}}}} & \left( {A\text{-}1} \right) \\{\mspace{79mu} {with}} & \; \\{\mspace{79mu} {\left( r_{i,j}^{S\; ɛ} \right)^{2} = {\left( r_{i}^{S\; ɛ} \right)^{2} + \left( r_{j}^{S\; ɛ} \right)^{2}}}} & \left( {A\text{-}2} \right)\end{matrix}$

the hypershere signature deviation (HSD) becomes

HSD_(i,j) D _(i,j) −r _(i,j) ^(Sε).   (26)

1. A computerized method for quantitative analysis of signal relatedmeasurements, the method comprising the steps of, performed with one ormore processors: producing an estimated signature typifying acharacteristic feature of the signal related measurements; computingmultidimensional statistics on the signal related measurements in amultidimensional space with respect to the estimated signature; andquantifying matching likelihoods of the signal related measurementsbased on distances of the signal related measurements with respect to ashell manifold derived from the multidimensional statistics andenveloping a signature manifold in the multidimensional space.
 2. Themethod according to claim 1, wherein the signature manifold is definedby an estimated signature which corresponds to: a point of themultidimensional space when the estimated signature is invariable withrespect to one or more conditions measured in the signal relatedmeasurements, in which case the shell manifold resembles a hypersphereshell; and a domain of the multidimensional space when the estimatedsignature changes with respect to one or more conditions measured in thesignal related measurements, in which case the shell manifold resemblesa structure enveloping the domain and having a dimension correspondingto a density probability function of a realization in themultidimensional space.
 3. The method according to claim 1, wherein themultidimensional statistics on the signal related measurements compriseshell-to-signature manifold average distance, standard deviation anddispersion of dispersion data with respect to the signature manifold inthe multidimensional space.
 4. The method according to claim 3, wherein,in the multidimensional statistics on the signal related measurementsand for a given condition set: the average distance data are defined as$r_{i} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}{\sqrt{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i.n}^{\prime}} \right)^{2}}\mspace{14mu} {or}}}}$$r_{i} = \sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i.n}^{\prime}} \right)^{2}}}}$for X_(m)∈ cluster i, where S′_(i,n) is a closest point of the estimatedsignature with respect to X_(m,n), N represents a number of dimensionsrelated to a representation form of the signal related measurements, Mrepresents a number of signal related measurements X, and S′ representsthe estimated signature produced in relation with a cluster i to whichthe signal related measurements belong; the standard deviation data aredefined as$\sigma_{i}^{r} = \sqrt{\frac{1}{M}{\sum\limits_{m = 1}^{M}\left( {r_{i} - \sqrt{\sum\limits_{n = 1}^{N}\left( {X_{m,n} - S_{i,n}} \right)^{2}}} \right)^{2}}}$or, for Gaussian noise assumption in the signal related measurements,${\sigma_{i}^{r} = {r_{i} \cdot \sqrt{\frac{1}{2\; N}}}};$ thedispersion of dispersion data are defined as$\left( \sigma_{i}^{\prime \; r} \right)^{2} = \frac{\sigma_{i}^{r\; 2}}{M - 1}$or, for Gaussian noise assumption in the signal related measurements,${\sigma_{i}^{\prime \; r} = {r_{i} \cdot \sqrt{\frac{1}{2{N \cdot \left( {M - 1} \right)}}}}};$and the multidimensional statistics on the signal related measurementsfurther comprise estimated average distance bias error data defined as${ɛ_{i}^{S} = {r_{i} \cdot \sqrt{\frac{1}{M\left( {M - 1} \right)}}}},{M > 1.}$5. The method according to claim 1, wherein the estimated signature isestimated from the signal related measurements using an averagingfunction.
 6. The method according to claim 5, wherein the estimatedsignature is estimated from the signal related measurements using amoving averaging function, the method further comprising the steps of:comparing a last one of the signal related measurements with acorresponding moving average resulting from the moving averagingfunction; comparing the moving average with a predetermined referencesignature; and determining a measurement trending based on comparisonresults from the steps of comparing.
 7. The method according to claim 6,further comprising the steps of: computing an average distance deviationcorresponding to a distance between said last one of the signal relatedmeasurements and the shell manifold; computing a total dispersion of thesignal related measurements; computing a density probability functionbased on the average distance deviation with respect to the totaldispersion, the measurement trending being determined according to thedensity probability function.
 8. The method according to claim 7,further comprising the step of producing a warning signal when apredetermined feature of said last one of the signal relatedmeasurements exceeds k times a standard deviation of measurementsdispersion around the shell manifold.
 9. The method according to claim1, wherein the estimated signature has evolution steps according to thesignal related measurements from which the estimated signature isderived, the method further comprising the steps of: computingmultidimensional statistics on the estimated signature at the evolutionsteps in the multidimensional space; and comparing the estimatedsignature at different ones of the evolution steps using themultidimensional statistics on the estimated signature to determine atrend.
 10. The method according to claim 9, further comprising the stepsof: determining a signature trending from the multidimensionalstatistics on the estimated signature; and when the signature trendinghas a deviation exceeding a preset threshold, performing patternrecognitions for the estimated signature with respect to a set ofsignatures stored in a database and indicative of predeterminedconditions.
 11. The method according to claim 9, wherein themultidimensional statistics on the estimated signature at the evolutionsteps comprise average distance, dispersion and internal error data. 12.The method according to claim 11, further comprising the steps of:computing a signature deviation based on a distance between theestimated signature at two of said evolution steps with respect to asignature shell manifold resulting from multidimensional statistics ondifferences of the estimated signature at different couples of saidevolution steps; computing a density probability function based on thesignature deviation; and determining a signature trending according tothe density probability function.
 13. The method according to claim 12,further comprising the step of producing a warning signal when thesignature deviation with respect to a signature-to-signature totaldispersion exceeds a preset signature deviation value.
 14. The methodaccording to claim 12, further comprising the step of merging aprobability of L signature-to-signature comparisons using a k factordefined as$k^{2} = {{\frac{1}{L}{\sum\limits_{l = 1}^{L}\left( \frac{{HSD}_{l}}{\sigma_{l}^{S - S}} \right)^{2}}} = {\frac{1}{L}{\sum\limits_{l = 1}^{L}k_{l}^{2}}}}$where HSD_(l) represents the signature deviation.
 15. The methodaccording to claim 1, wherein the signal related measurements are formedof signal measurements normalized as function of a predetermined noiseamplitude normalization metric.
 16. The method according to claim 1,further comprising the steps of: producing a set of estimated signaturestypifying characteristic features of sets of the signal relatedmeasurements; and determining membership likelihoods of the signalrelated measurements for the set of estimated signatures based ondistances of the estimated signatures with respect to shell manifoldsderived from the multidimensional statistics and located around thesignal related measurements.
 17. The method according to claim 1,wherein the signal related measurements are formed of averages of signalmeasurements split into sets.
 18. The method according to claim 1,wherein the estimated signature is estimated from signal measurements inrespect with a first apparatus similar to a second apparatus from whichthe signal related measurements derive and having similar operatingconditions.
 19. The method according to claim 1, further comprising thesteps of: determining a measurement trending from the multidimensionalstatistics on the signal related measurements; and when the measurementtrending has a deviation to the estimated signature exceeding a presetthreshold, performing pattern recognitions for the estimated signaturewith respect to a set of signatures stored in a database and indicativeof predetermined conditions.
 20. The method according to claim 1,further comprising the steps of: estimating a covariance matrix with thesignal related measurements; computing multidimensional statistics onthe covariance matrix in a multidimensional space of N×N dimensionswhere N is a dimensionality of the estimated signature; and quantifyingmatching likelihoods of the signal related measurements also based ondistances of cross-measurement matrices with respect to the estimatedcovariance matrix.
 21. The method according to claim 1, furthercomprising the step of krigging the signal related measurements to buildthe signature manifold.
 22. A system for monitoring an operatingcondition of an apparatus, comprising: a measuring arrangementconnectable to the apparatus, the measuring arrangement being configuredto measure one or more predetermined operating parameters of theapparatus and produce signal related measurements thereof; a memoryhaving a statistics database; a processor connected to the measuringarrangement and the memory, the processor being configured to processthe signal related measurements, produce multidimensional statistics onthe signal related measurements and an estimated signature typifying acharacteristic feature of the signal related measurements, updating thestatistics database with the signal related measurements and themultidimensional statistics, and produce diagnosis data indicative ofthe operating condition of the apparatus as function of matchinglikelihoods of the signal related measurements quantified based ondistances of the signal related measurements with respect to a shellmanifold derived from the multidimensional statistics and enveloping asignature manifold in the multidimensional space; and an output unitconnected to the processor for externally reporting the diagnosis data.23. The system according to claim 22, further comprising a communicationunit connected to the processor and the memory and connectable to acommunication link with at least one like system monitoring a likeapparatus for exchanging data therewith.
 24. The system according toclaim 22, further comprising a control unit connected to the processorand the measuring arrangement and connectable to the apparatus forproducing control signals for the apparatus as function of control dataproduced by the processor based on the diagnosis data.
 25. The systemaccording to claim 22, wherein the processor has a signature statisticsmodule computing the multidimensional statistics on the estimatedsignature from the signal related measurements, a measurement statisticsmodule computing the multidimensional statistics on the signal relatedmeasurements from the signal related measurements, a measurementtrending module performing measurement trending from themultidimensional statistics computed by the signature and measurementstatistics modules, and a statistics database management module managingthe statistics database from the multidimensional statistics computed bythe signature and measurement statistics modules.
 26. The systemaccording to claim 25, wherein the processor further has a measurementtrending or pattern recognition module performing measurement trendingor pattern recognition from the multidimensional statistics computed bythe measurement statistics module and the multidimensional statistics onthe estimated signature from the statistic database management module,and a signature trending or pattern recognition module performingsignature trending or pattern recognition from the multidimensionalstatistics computed by the signature statistics module and themultidimensional statistics on the estimated signature from thestatistics database management module.
 27. The system according to claim26, wherein the diagnosis data comprise alarm state data derived fromthe measurement trending, the measurement trending or patternrecognition and the signature trending or pattern recognition modules,indicative of an abnormal operating state of the apparatus.